NONAUTONOMOUS EVOLUTIONARY EQUATIONS WITH TRANSLATION-COMPACT SYMBOLSAND THEIR ATTRACTORS

We study a new class of non-autonomous evolutionary equations arising in mathematical physics. These equations contain time-dependent non-li near functions and right-hand sides (i.e. time symbols of equations) w hich satisfy the translation-compact (tr.c.) property. The latter mean s that the set of all time translations of such functions forms a prec ompact set with respect to the corresponding functional space topology . The class of symbols under consideration is much wider than the clas s of almost periodic (a.p.) symbol studied in [1]. We estblish the exi stence and describe the structure of attractors for the equations with tr.c. symbols. For example, for 2 D Navier-Stokes (NS) system with an external force phi(x, t) such that integral(t+1)(t) parallel to phi(. , s)parallel to(H)(2) ds less than or equal to M < +infinity, For All t is an element of R, we prove the existence of the uniform attractor and describe its structure. Similar results are established for the wi de class of non-autonomous reaction-diffusion systems and other equati ons.